Functional inverse of $\sin\theta\sqrt{\tan\theta}$

Question

What is the functional inverse of $f(\theta) = \sin\theta\sqrt{\tan\theta}$? Or, equivalently, what is the inverse of $$f(\theta)=\sin^2\,\theta\tan\,\theta=\frac{\sin^3\,\theta}{\cos\,\theta}$$

It comes from a physics setup involving two equivalently massed and charged pith balls separated by a certain distance, and the equation simplifies to $q = 4L\sin\theta\sqrt{\pi\epsilon_0mg\tan{\theta}}$, where $\pi$, $g$, and $\epsilon_0$ are the obvious physical constants and $L$ and $m$ will be fixed. The question asked for $\theta$ in terms of $q$, however, so I'm wondering if there is a way to rearrange this. I can't seem to find anything on the internet, and Wolfram refuses to reveal the steps for its complex rearranged formula.

Answer 1

I will assume you are interested in finding $\theta = f^{-1}(x)$ for $x \geq 0$ with the range $0 \leq \theta < \frac{\pi}{2}$. $$ x^2 = \left(f(\theta)\right)^2 = \sin^2(\theta) \tan(\theta) = \frac{\tan^3(\theta)}{1+\tan^2(\theta)} $$ Hence $\theta = \arctan(y(x))$, where $y$ is the positive root of $y^3 = x^2 (1 + y^2)$.

Using Cardano's formula: $$ \theta(x) = \arctan\left( \frac{1}{3} \left(x^2+\frac{\sqrt[3]{2} x^{10/3}}{\sqrt[3]{2 x^4+3 \left(\sqrt{12 x^4+81}+9\right)}}+\frac{\sqrt[3]{2 x^4+3 \left(\sqrt{12 x^4+81}+9\right)} x^{2/3}}{\sqrt[3]{2}}\right) \right) $$